View
293
Download
3
Category
Preview:
Citation preview
PYP110Printed in the United Kingdom by Antony Rowe Ltd, Chippenham, Wiltshire
Published February 2009
International BaccalaureatePeterson House, Malthouse Avenue, Cardiff Gate
Cardiff, Wales GB CF23 8GLUnited Kingdom
Phone: +44 29 2054 7777Fax: +44 29 2054 7778
Website: http://www.ibo.org
© International Baccalaureate Organization 2009
The International Baccalaureate (IB) offers three high quality and challenging educational programmes for a worldwide community of schools, aiming to create a better, more peaceful world.
The IB is grateful for permission to reproduce and/or translate any copyright material used in this publication. Acknowledgments are included, where appropriate, and, if notified, the IB will be pleased to rectify any errors or omissions at the earliest opportunity.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior written permission of the IB, or as expressly permitted by law or by the IB’s own rules and policy. See http://www.ibo.org/copyright.
IB merchandise and publications can be purchased through the IB store at http://store.ibo.org. General ordering queries should be directed to the sales and marketing department in Cardiff.
Phone: +44 29 2054 7746Fax: +44 29 2054 7779Email: sales@ibo.org
Primary Years Programme
Mathematics scope and sequence
IB mission statementThe International Baccalaureate aims to develop inquiring, knowledgeable and caring young people who help to create a better and more peaceful world through intercultural understanding and respect.
To this end the organization works with schools, governments and international organizations to develop challenging programmes of international education and rigorous assessment.
These programmes encourage students across the world to become active, compassionate and lifelong learners who understand that other people, with their differences, can also be right.
IB learner profileThe aim of all IB programmes is to develop internationally minded people who, recognizing their common humanity and shared guardianship of the planet, help to create a better and more peaceful world.
IB learners strive to be:
Inquirers They develop their natural curiosity. They acquire the skills necessary to conduct inquiry and research and show independence in learning. They actively enjoy learning and this love of learning will be sustained throughout their lives.
Knowledgeable They explore concepts, ideas and issues that have local and global significance. In so doing, they acquire in-depth knowledge and develop understanding across a broad and balanced range of disciplines.
Thinkers They exercise initiative in applying thinking skills critically and creatively to recognize and approach complex problems, and make reasoned, ethical decisions.
Communicators They understand and express ideas and information confidently and creatively in more than one language and in a variety of modes of communication. They work effectively and willingly in collaboration with others.
Principled They act with integrity and honesty, with a strong sense of fairness, justice and respect for the dignity of the individual, groups and communities. They take responsibility for their own actions and the consequences that accompany them.
Open-minded They understand and appreciate their own cultures and personal histories, and are open to the perspectives, values and traditions of other individuals and communities. They are accustomed to seeking and evaluating a range of points of view, and are willing to grow from the experience.
Caring They show empathy, compassion and respect towards the needs and feelings of others. They have a personal commitment to service, and act to make a positive difference to the lives of others and to the environment.
Risk-takers They approach unfamiliar situations and uncertainty with courage and forethought, and have the independence of spirit to explore new roles, ideas and strategies. They are brave and articulate in defending their beliefs.
Balanced They understand the importance of intellectual, physical and emotional balance to achieve personal well-being for themselves and others.
Reflective They give thoughtful consideration to their own learning and experience. They are able to assess and understand their strengths and limitations in order to support their learning and personal development.
© International Baccalaureate Organization 2007
Mathematics scope and sequence
Contents
Introduction to the PYP mathematics scope and sequence 1
What the PYP believes about learning mathematics 1
Mathematics in a transdisciplinary programme 2
The structure of the PYP mathematics scope and sequence 3
How to use the PYP mathematics scope and sequence 4
Viewing a unit of inquiry through the lens of mathematics 5
Learning continuums 6
Data handling 6
Measurement 10
Shape and space 14
Pattern and function 18
Number 21
Samples 26
Mathematics scope and sequence 1
Introduction to the PYP mathematics scope and sequence
The information in this scope and sequence document should be read in conjunction with the mathematics subject annex in Making the PYP happen: A curriculum framework for international primary education (2007).
What the PYP believes about learning mathematicsThe power of mathematics for describing and analysing the world around us is such that it has become a highly effective tool for solving problems. It is also recognized that students can appreciate the intrinsic fascination of mathematics and explore the world through its unique perceptions. In the same way that students describe themselves as “authors” or “artists”, a school’s programme should also provide students with the opportunity to see themselves as “mathematicians”, where they enjoy and are enthusiastic when exploring and learning about mathematics.
In the IB Primary Years Programme (PYP), mathematics is also viewed as a vehicle to support inquiry, providing a global language through which we make sense of the world around us. It is intended that students become competent users of the language of mathematics, and can begin to use it as a way of thinking, as opposed to seeing it as a series of facts and equations to be memorized.
How children learn mathematicsIt is important that learners acquire mathematical understanding by constructing their own meaning through ever-increasing levels of abstraction, starting with exploring their own personal experiences, understandings and knowledge. Additionally, it is fundamental to the philosophy of the PYP that, since it is to be used in real-life situations, mathematics needs to be taught in relevant, realistic contexts, rather than by attempting to impart a fixed body of knowledge directly to students. How children learn mathematics can be described using the following stages (see figure 1).
Applying with understanding
Transferring meaning
Constructing meaning
Figure 1How children learn mathematics
Introduction to the PYP mathematics scope and sequence
Mathematics scope and sequence2
Constructing meaning about mathematicsLearners construct meaning based on their previous experiences and understanding, and by reflecting upon their interactions with objects and ideas. Therefore, involving learners in an active learning process, where they are provided with possibilities to interact with manipulatives and to engage in conversations with others, is paramount to this stage of learning mathematics.
When making sense of new ideas all learners either interpret these ideas to conform to their present understanding or they generate a new understanding that accounts for what they perceive to be occurring. This construct will continue to evolve as learners experience new situations and ideas, have an opportunity to reflect on their understandings and make connections about their learning.
Transferring meaning into symbolsOnly when learners have constructed their ideas about a mathematical concept should they attempt to transfer this understanding into symbols. Symbolic notation can take the form of pictures, diagrams, modelling with concrete objects and mathematical notation. Learners should be given the opportunity to describe their understanding using their own method of symbolic notation, then learning to transfer them into conventional mathematical notation.
Applying with understandingApplying with understanding can be viewed as the learners demonstrating and acting on their understanding. Through authentic activities, learners should independently select and use appropriate symbolic notation to process and record their thinking. These authentic activities should include a range of practical hands-on problem-solving activities and realistic situations that provide the opportunity to demonstrate mathematical thinking through presented or recorded formats. In this way, learners are able to apply their understanding of mathematical concepts as well as utilize mathematical skills and knowledge.
As they work through these stages of learning, students and teachers use certain processes of mathematical reasoning.
They use patterns and relationships to analyse the problem situations upon which they are working.•
They make and evaluate their own and each other’s ideas.•
They use models, facts, properties and relationships to explain their thinking.•
They justify their answers and the processes by which they arrive at solutions.•
In this way, students validate the meaning they construct from their experiences with mathematical situations. By explaining their ideas, theories and results, both orally and in writing, they invite constructive feedback and also lay out alternative models of thinking for the class. Consequently, all benefit from this interactive process.
Mathematics in a transdisciplinary programmeWherever possible, mathematics should be taught through the relevant, realistic context of the units of inquiry. The direct teaching of mathematics in a unit of inquiry may not always be feasible but, where appropriate, prior learning or follow-up activities may be useful to help students make connections between the different aspects of the curriculum. Students also need opportunities to identify and reflect on “big ideas” within and between the different strands of mathematics, the programme of inquiry and other subject areas.
Links to the transdisciplinary themes should be explicitly made, whether or not the mathematics is being taught within the programme of inquiry. A developing understanding of these links will contribute to the students’ understanding of mathematics in the world and to their understanding of the transdisciplinary
Introduction to the PYP mathematics scope and sequence
Mathematics scope and sequence 3
theme. The role of inquiry in mathematics is important, regardless of whether it is being taught inside or outside the programme of inquiry. However, it should also be recognized that there are occasions when it is preferable for students to be given a series of strategies for learning mathematical skills in order to progress in their mathematical understanding rather than struggling to proceed.
The structure of the PYP mathematics scope and sequenceThis scope and sequence aims to provide information for the whole school community of the learning that is going on in the subject area of mathematics. It has been designed in recognition that learning mathematics is a developmental process and that the phases a learner passes through are not always linear or age related. For this reason the content is presented in continuums for each of the five strands of mathematics—data handling, measurement, shape and space, pattern and function, and number. For each of the strands there is a strand description and a set of overall expectations. The overall expectations provide a summary of the understandings and subsequent learning being developed for each phase within a strand.
The content of each continuum has been organized into four phases of development, with each phase building upon and complementing the previous phase. The continuums make explicit the conceptual understandings that need to be developed at each phase. Evidence of these understandings is described in the behaviours or learning outcomes associated with each phase and these learning outcomes relate specifically to mathematical concepts, knowledge and skills.
The learning outcomes have been written to reflect the stages a learner goes through when developing conceptual understanding in mathematics—constructing meaning, transferring meaning into symbols and applying with understanding (see figure 1). To begin with, the learning outcomes identified in the constructing meaning stage strongly emphasize the need for students to develop understanding of mathematical concepts in order to provide them with a secure base for further learning. In the planning process, teachers will need to discuss the ways in which students may demonstrate this understanding. The amount of time and experiences dedicated to this stage of learning will vary from student to student.
The learning outcomes in the transferring meaning into symbols stage are more obviously demonstrable and observable. The expectation for students working in this stage is that they have demonstrated understanding of the underlying concepts before being asked to transfer this meaning into symbols. It is acknowledged that, in some strands, symbolic representation will form part of the constructing meaning stage. For example, it is difficult to imagine how a student could construct meaning about the way in which information is expressed as organized and structured data without having the opportunity to collect and represent this data in graphs. In this type of example, perhaps the difference between the two stages is that in the transferring meaning into symbols stage the student will be able to demonstrate increased independence with decreasing amounts of teacher prompting required for them to make connections. Another difference could be that a student’s own symbolic representation may be extended to include more conventional methods of symbolic representation.
In the final stage, a number of learning outcomes have been developed to reflect the kind of actions and behaviours that students might demonstrate when applying with understanding. It is important to note that other forms of application might be in evidence in classrooms where there are authentic opportunities for students to make spontaneous connections between the learning that is going on in mathematics and other areas of the curriculum and daily life.
When a continuum for a particular strand is observed as a whole, it is clear how the conceptual understandings and the associated learning outcomes develop in complexity as they are viewed across the phases. In each of the phases, there is also a vertical progression where most learning outcomes identified in the constructing meaning stage of the phase are often described as outcomes relating to the transferring
Introduction to the PYP mathematics scope and sequence
Mathematics scope and sequence4
meaning into symbols and applying with understanding stages of the same phase. However, on some occasions, a mathematical concept is introduced in one phase but students are not expected to apply the concept until a later phase. This is a deliberate decision aimed at providing students with adequate time and opportunities for the ongoing development of understanding of particular concepts.
Each of the continuums contains a notes section which provides extra information to clarify certain learning outcomes and to support planning, teaching and learning of particular concepts.
How to use the PYP mathematics scope and sequenceIn the course of reviewing the PYP mathematics scope and sequence, a decision was made to provide schools with a view of how students construct meaning about mathematics concepts in a more developmental way rather than in fixed age bands. Teachers will need to be given time to discuss this introduction and accompanying continuums and how they can be used to inform planning, teaching and assessing of mathematics in the school. The following points should also be considered in this discussion process.
It is acknowledged that there are earlier and later phases that have not been described in these •continuums.
Each learner is a unique individual with different life experiences and no two learning pathways are •the same.
Learners within the same age group will have different proficiency levels and needs; therefore, teachers •should consider a range of phases when planning mathematics learning experiences for a class.
Learners are likely to display understanding and skills from more than one of the phases at a time. •Consequently, it is recognized that teachers will interpret this scope and sequence according to the needs of their students and their particular teaching situations.
The continuums are not prescriptive tools that assume a learner must attain all the outcomes of a •particular phase before moving on to the next phase, nor that the learner should be in the same phase for each strand.
Each teacher needs to identify the extent to which these factors affect the learner. Plotting a mathematical profile for each student is a complex process for PYP teachers. Prior knowledge should therefore never be assumed before embarking on the presentation or introduction of a mathematical concept.
Schools may decide to use and adapt the PYP scope and sequences according to their needs. For example, a school may decide to frame their mathematics scope and sequence document around the conceptual understandings outlined in the PYP document, but develop other aspects (for example outcomes, indicators, benchmarks, standards) differently. Alternatively, they may decide to incorporate the continuums from the PYP documents into their existing school documents. Schools need to be mindful of practice C1.23 in the IB Programme standards and practices (2005) that states, “If the school adapts, or develops, its own scope and sequence documents for each PYP subject area, the level of overall expectation regarding student achievement expressed in these documents at least matches that expressed in the PYP scope and sequence documents”. To arrive at such a judgment, and given that the overall expectations in the PYP mathematics scope and sequence are presented as broad generalities, it is recommended that the entire document be read and considered.
Introduction to the PYP mathematics scope and sequence
Mathematics scope and sequence 5
Viewing a unit of inquiry through the lens of mathematicsThe following diagram shows a sample process for viewing a unit of inquiry through the lens of mathematics. This has been developed as an example of how teachers can identify the mathematical concepts, skills and knowledge required to successfully engage in the units of inquiry.
Note: It is important that the integrity of a central idea and ensuing inquiry is not jeopardized by a subject-specific focus too early in the collaborative planning process. Once an inquiry has been planned through to identification of learning experiences, it would be appropriate to consider the following process.
Figure 2Sample processes for viewing a unit of inquiry through the lens of mathematics
Will mathematics inform this unit?
Do aspects of the transdisciplinary theme initially stand out as being mathematics related?
Will mathematical knowledge, concepts and skills be needed to understand the central idea?
Will mathematical knowledge, concepts and skills be needed to develop the lines of inquiry within the unit?
What mathematical knowledge, concepts and skills will the students need to be able to engage with and/or inquire into the following? (Refer to mathematics scope and sequence documents.)
Central idea•
Lines of inquiry•
Assessment tasks•
Teacher questions, student questions•
Learning experiences•
In your planning team, list the knowledge, concepts and skills.
What prior knowledge, concepts and skills do the students have that can be utilized and built upon?
Which stages of understanding are the learners in the class working at—constructing meaning, transferring meaning into symbols, or applying with understanding?
How will we know what they have learned? Identify opportunities for assessment.
Decide which aspects can be learned:
within the unit of inquiry (learning through mathematics)•
as subject-specific, prior to being used and applied in the context of the inquiry (inquiry into •mathematics).
Mathematics scope and sequence6
Learning continuums
Data handlingData handling allows us to make a summary of what we know about the world and to make inferences about what we do not know.
Data can be collected, organized, represented and summarized in a variety of ways to highlight •similarities, differences and trends; the chosen format should illustrate the information without bias or distortion.
Probability can be expressed qualitatively by using terms such as “unlikely”, “certain” or “impossible”. It •can be expressed quantitatively on a numerical scale.
Overall expectationsPhase 1Learners will develop an understanding of how the collection and organization of information helps to make sense of the world. They will sort, describe and label objects by attributes and represent information in graphs including pictographs and tally marks. The learners will discuss chance in daily events.
Phase 2Learners will understand how information can be expressed as organized and structured data and that this can occur in a range of ways. They will collect and represent data in different types of graphs, interpreting the resulting information for the purpose of answering questions. The learners will develop an understanding that some events in daily life are more likely to happen than others and they will identify and describe likelihood using appropriate vocabulary.
Phase 3Learners will continue to collect, organize, display and analyse data, developing an understanding of how different graphs highlight different aspects of data more efficiently. They will understand that scale can represent different quantities in graphs and that mode can be used to summarize a set of data. The learners will make the connection that probability is based on experimental events and can be expressed numerically.
Phase 4Learners will collect, organize and display data for the purposes of valid interpretation and communication. They will be able to use the mode, median, mean and range to summarize a set of data. They will create and manipulate an electronic database for their own purposes, including setting up spreadsheets and using simple formulas to create graphs. Learners will understand that probability can be expressed on a scale (0–1 or 0%–100%) and that the probability of an event can be predicted theoretically.
Learning continuums
Mathematics scope and sequence 7
Lear
nin
g c
on
tin
uu
m fo
r d
ata
han
dlin
g
Ph
ase
1 P
has
e 2
Ph
ase
3 P
has
e 4
Co
nce
ptu
al u
nd
erst
and
ing
sW
e co
llect
info
rmat
ion
to m
ake
sens
e of
th
e w
orld
aro
und
us.
Org
aniz
ing
obje
cts
and
even
ts h
elp
s us
to
solv
e p
rob
lem
s.
Even
ts in
dai
ly li
fe in
volv
e ch
ance
.
Co
nce
ptu
al u
nd
erst
and
ing
sIn
form
atio
n ca
n b
e ex
pre
ssed
as
orga
nize
d an
d st
ruct
ured
dat
a.
Ob
ject
s an
d ev
ents
can
be
orga
nize
d in
d
iffe
rent
way
s.
Som
e ev
ents
in d
aily
life
are
mor
e lik
ely
to
hap
pen
than
oth
ers.
Co
nce
ptu
al u
nd
erst
and
ing
sD
ata
can
be
colle
cted
, org
aniz
ed,
dis
pla
yed
and
anal
ysed
in d
iffe
rent
way
s.
Dif
fere
nt g
rap
h fo
rms
high
light
dif
fere
nt
asp
ects
of d
ata
mor
e ef
ficie
ntly
.
Prob
abili
ty c
an b
e b
ased
on
exp
erim
enta
l ev
ents
in d
aily
life
.
Prob
abili
ty c
an b
e ex
pre
ssed
in n
umer
ical
no
tati
ons.
Co
nce
ptu
al u
nd
erst
and
ing
sD
ata
can
be
pre
sent
ed e
ffec
tive
ly fo
r val
id
inte
rpre
tati
on a
nd c
omm
unic
atio
n.
Rang
e, m
ode,
med
ian
and
mea
n ca
n b
e us
ed to
ana
lyse
sta
tisti
cal d
ata.
Prob
abili
ty c
an b
e re
pre
sent
ed o
n a
scal
e b
etw
een
0–1
or 0
%–1
00%
.
The
pro
bab
ility
of a
n ev
ent c
an b
e p
red
icte
d th
eore
tica
lly.
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
that
set
s ca
n b
e or
gani
zed
•b
y d
iffe
rent
att
rib
utes
und
erst
and
that
info
rmat
ion
abou
t •
them
selv
es a
nd th
eir s
urro
und
ings
ca
n b
e ob
tain
ed in
dif
fere
nt w
ays
dis
cuss
cha
nce
in d
aily
eve
nts
•(im
pos
sib
le, m
ayb
e, c
erta
in).
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
that
set
s ca
n b
e or
gani
zed
•b
y on
e or
mor
e at
trib
utes
und
erst
and
that
info
rmat
ion
abou
t •
them
selv
es a
nd th
eir s
urro
und
ings
ca
n b
e co
llect
ed a
nd re
cord
ed in
d
iffe
rent
way
s
und
erst
and
the
conc
ept o
f cha
nce
in
•d
aily
eve
nts
(imp
ossi
ble
, les
s lik
ely,
m
ayb
e, m
ost l
ikel
y, c
erta
in).
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
that
dat
a ca
n b
e •
colle
cted
, dis
pla
yed
and
inte
rpre
ted
usin
g si
mp
le g
rap
hs, f
or e
xam
ple
, bar
gr
aphs
, lin
e gr
aphs
und
erst
and
that
sca
le c
an re
pre
sent
•
dif
fere
nt q
uant
itie
s in
gra
phs
und
erst
and
that
the
mod
e ca
n b
e •
used
to s
umm
ariz
e a
set o
f dat
a
und
erst
and
that
one
of t
he p
urp
oses
•
of a
dat
abas
e is
to a
nsw
er q
uest
ions
an
d so
lve
pro
ble
ms
und
erst
and
that
pro
bab
ility
is b
ased
•
on e
xper
imen
tal e
vent
s.
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
that
dif
fere
nt t
ypes
of
•gr
aphs
hav
e sp
ecia
l pur
pos
es
und
erst
and
that
the
mod
e, m
edia
n,
•m
ean
and
rang
e ca
n su
mm
ariz
e a
set
of d
ata
und
erst
and
that
pro
bab
ility
can
be
•ex
pre
ssed
in s
cale
(0–1
) or p
er c
ent
(0%
–100
%)
und
erst
and
the
dif
fere
nce
bet
wee
n •
exp
erim
enta
l and
theo
reti
cal
pro
bab
ility
.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
rep
rese
nt in
form
atio
n th
roug
h •
pic
togr
aphs
and
tally
mar
ks
sort
and
lab
el re
al o
bje
cts
by
•at
trib
utes
.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
colle
ct a
nd re
pre
sent
dat
a in
dif
fere
nt
•ty
pes
of g
rap
hs, f
or e
xam
ple
, tal
ly
mar
ks, b
ar g
rap
hs
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
colle
ct, d
isp
lay
and
inte
rpre
t dat
a •
usin
g si
mp
le g
rap
hs, f
or e
xam
ple
, bar
gr
aphs
, lin
e gr
aphs
iden
tify
, rea
d an
d in
terp
ret r
ang
e an
d •
scal
e on
gra
phs
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
colle
ct, d
ispl
ay a
nd in
terp
ret d
ata
in
•ci
rcle
gra
phs
(pie
cha
rts)
and
line
gra
phs
iden
tify
, des
crib
e an
d ex
pla
in th
e •
rang
e, m
ode,
med
ian
and
mea
n in
a
set o
f dat
a
Learning continuums
Mathematics scope and sequence8
repr
esen
t the
rela
tions
hip
bet
wee
n •
obje
cts
in s
ets
usin
g tr
ee, V
enn
and
Car
roll
diag
ram
s
exp
ress
the
chan
ce o
f an
even
t •
hap
pen
ing
usin
g w
ords
or p
hras
es
(imp
ossi
ble
, les
s lik
ely,
may
be,
mos
t lik
ely,
cer
tain
).
iden
tify
the
mod
e of
a s
et o
f dat
a•
use
tree
dia
gram
s to
exp
ress
•
pro
bab
ility
usi
ng s
imp
le fr
acti
ons.
set u
p a
sp
read
shee
t usi
ng s
imp
le
•fo
rmul
as to
man
ipul
ate
dat
a an
d to
cr
eate
gra
phs
exp
ress
pro
bab
iliti
es u
sing
sca
le (0
–1)
•or
per
cen
t (0%
–100
%).
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
crea
te p
icto
grap
hs a
nd ta
lly m
arks
•
crea
te li
ving
gra
phs
usi
ng re
al o
bje
cts
•an
d p
eop
le*
des
crib
e re
al o
bje
cts
and
even
ts b
y •
attr
ibut
es.
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
colle
ct, d
isp
lay
and
inte
rpre
t dat
a fo
r •
the
pur
pos
e of
ans
wer
ing
ques
tion
s
crea
te a
pic
togr
aph
and
sam
ple
bar
•
grap
h of
real
ob
ject
s an
d in
terp
ret
dat
a b
y co
mp
arin
g qu
anti
ties
(for
ex
amp
le, m
ore,
few
er, l
ess
than
, gr
eate
r tha
n)
use
tree
, Ven
n an
d C
arro
ll d
iagr
ams
to
•ex
plo
re re
lati
onsh
ips
bet
wee
n d
ata
iden
tify
and
des
crib
e ch
ance
in d
aily
•
even
ts (i
mp
ossi
ble
, les
s lik
ely,
may
be,
m
ost l
ikel
y, c
erta
in).
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
des
ign
a su
rvey
and
sys
tem
atic
ally
•
colle
ct, o
rgan
ize
and
dis
pla
y d
ata
in
pic
togr
aphs
and
bar
gra
phs
sele
ct a
pp
rop
riat
e gr
aph
form
(s) t
o •
dis
pla
y d
ata
inte
rpre
t ran
ge
and
scal
e on
gra
phs
•
use
pro
bab
ility
to d
eter
min
e •
mat
hem
atic
ally
fair
and
unfa
ir ga
mes
an
d to
exp
lain
pos
sib
le o
utco
mes
exp
ress
pro
bab
ility
usi
ng s
imp
le
•fr
acti
ons.
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
des
ign
a su
rvey
and
sys
tem
atic
ally
•
colle
ct, r
ecor
d, o
rgan
ize
and
dis
pla
y th
e d
ata
in a
bar
gra
ph,
circ
le g
rap
h,
line
grap
h
iden
tify
, des
crib
e an
d ex
pla
in th
e •
rang
e, m
ode,
med
ian
and
mea
n in
a
set o
f dat
a
crea
te a
nd m
anip
ulat
e an
ele
ctro
nic
•d
atab
ase
for t
heir
own
pur
pos
es
det
erm
ine
the
theo
reti
cal p
rob
abili
ty
•of
an
even
t and
exp
lain
why
it m
ight
d
iffe
r fro
m e
xper
imen
tal p
rob
abili
ty.
No
tes
Uni
ts o
f inq
uiry
will
be
rich
in
opp
ortu
niti
es fo
r col
lect
ing
and
orga
nizi
ng in
form
atio
n. It
may
be
usef
ul
for t
he te
ache
r to
pro
vid
e sc
affo
lds,
suc
h as
que
stio
ns fo
r exp
lora
tion
, and
the
mod
ellin
g of
gra
phs
and
dia
gram
s.
*Liv
ing
grap
hs re
fer t
o d
ata
that
is
orga
nize
d b
y p
hysi
cally
mov
ing
and
arra
ngin
g st
uden
ts o
r act
ual m
ater
ials
in
suc
h a
way
as
to s
how
and
com
par
e qu
anti
ties
.
No
tes
An
incr
easi
ng n
umb
er o
f com
put
er a
nd
web
-bas
ed a
pp
licat
ions
are
ava
ilab
le th
at
enab
le le
arne
rs to
man
ipul
ate
dat
a in
or
der
to c
reat
e gr
aphs
.
Stud
ents
sho
uld
have
a lo
t of e
xper
ienc
e of
org
aniz
ing
dat
a in
a v
arie
ty o
f way
s,
and
of ta
lkin
g ab
out t
he a
dva
ntag
es a
nd
dis
adva
ntag
es o
f eac
h. In
terp
reta
tion
s of
d
ata
shou
ld in
clud
e th
e in
form
atio
n th
at
cann
ot b
e co
nclu
ded
as
wel
l as
that
whi
ch
can.
It is
imp
orta
nt to
rem
emb
er th
at
the
chos
en fo
rmat
sho
uld
illus
trat
e th
e in
form
atio
n w
itho
ut b
ias.
No
tes
Usi
ng d
ata
that
has
bee
n co
llect
ed a
nd
save
d is
a s
imp
le w
ay to
beg
in d
iscu
ssin
g th
e m
ode.
A fu
rthe
r ext
ensi
on o
f mod
e is
to
form
ulat
e th
eori
es a
bou
t why
a c
erta
in
choi
ce is
the
mod
e.
Stud
ents
sho
uld
have
the
opp
ortu
nity
to
use
dat
abas
es, i
dea
lly, t
hose
cre
ated
us
ing
dat
a co
llect
ed b
y th
e st
uden
ts th
en
ente
red
into
a d
atab
ase
by
the
teac
her o
r to
get
her.
No
tes
A d
atab
ase
is a
col
lect
ion
of d
ata,
whe
re
the
dat
a ca
n b
e d
isp
laye
d in
man
y fo
rms.
Th
e d
ata
can
be
chan
ged
at a
ny ti
me.
A
spre
adsh
eet i
s a
typ
e of
dat
abas
e w
here
in
form
atio
n is
set
out
in a
tab
le. U
sing
a
com
mon
set
of d
ata
is a
goo
d w
ay fo
r st
uden
ts to
sta
rt to
set
up
thei
r ow
n d
atab
ases
. A u
nit o
f inq
uiry
wou
ld b
e an
exc
elle
nt s
ourc
e of
com
mon
dat
a fo
r st
uden
t pra
ctic
e.
Learning continuums
Mathematics scope and sequence 9
Very
you
ng c
hild
ren
view
the
wor
ld a
s a
pla
ce o
f pos
sib
iliti
es. T
he te
ache
r sho
uld
try
to in
trod
uce
pra
ctic
al e
xam
ple
s an
d sh
ould
use
ap
pro
pri
ate
voca
bul
ary.
D
iscu
ssio
ns a
bou
t cha
nce
in d
aily
eve
nts
shou
ld b
e re
leva
nt to
the
cont
ext o
f the
le
arne
rs.
Situ
atio
ns th
at c
ome
up n
atur
ally
in
the
clas
sroo
m, o
ften
thro
ugh
liter
atur
e,
pre
sent
op
por
tuni
ties
for d
iscu
ssin
g p
rob
abili
ty. D
iscu
ssio
ns n
eed
to ta
ke
pla
ce in
whi
ch s
tud
ents
can
sha
re th
eir
sens
e of
like
lihoo
d in
term
s th
at a
re u
sefu
l to
them
.
Situ
atio
ns th
at c
ome
up n
atur
ally
in th
e cl
assr
oom
or f
orm
par
t of t
he u
nits
of
inqu
iry
pre
sent
op
por
tuni
ties
for s
tud
ents
to
furt
her d
evel
op th
eir u
nder
stan
din
g of
st
atis
tics
and
pro
bab
ility
con
cep
ts.
Tech
nolo
gy
give
s us
the
opti
on o
f cr
eati
ng a
gra
ph
at th
e p
ress
of a
key
. Be
ing
able
to g
ener
ate
dif
fere
nt t
ypes
of
gra
phs
allo
ws
lear
ners
to e
xplo
re a
nd
app
reci
ate
the
attr
ibut
es o
f eac
h ty
pe
of
grap
h an
d it
s ef
ficac
y in
dis
pla
ying
the
dat
a.
Tech
nolo
gy
also
giv
es u
s th
e p
ossi
bili
ty
of ra
pid
ly re
plic
atin
g ra
ndom
eve
nts.
C
omp
uter
and
web
-bas
ed a
pp
licat
ions
ca
n b
e us
ed to
toss
coi
ns, r
oll d
ice,
and
ta
bul
ate
and
grap
h th
e re
sult
s.
Learning continuums
Mathematics scope and sequence10
MeasurementTo measure is to attach a number to a quantity using a chosen unit. Since the attributes being measured are continuous, ways must be found to deal with quantities that fall between numbers. It is important to know how accurate a measurement needs to be or can ever be.
Overall expectationsPhase 1Learners will develop an understanding of how measurement involves the comparison of objects and the ordering and sequencing of events. They will be able to identify, compare and describe attributes of real objects as well as describe and sequence familiar events in their daily routine.
Phase 2Learners will understand that standard units allow us to have a common language to measure and describe objects and events, and that while estimation is a strategy that can be applied for approximate measurements, particular tools allow us to measure and describe attributes of objects and events with more accuracy. Learners will develop these understandings in relation to measurement involving length, mass, capacity, money, temperature and time.
Phase 3Learners will continue to use standard units to measure objects, in particular developing their understanding of measuring perimeter, area and volume. They will select and use appropriate tools and units of measurement, and will be able to describe measures that fall between two numbers on a scale. The learners will be given the opportunity to construct meaning about the concept of an angle as a measure of rotation.
Phase 4Learners will understand that a range of procedures exists to measure different attributes of objects and events, for example, the use of formulas for finding area, perimeter and volume. They will be able to decide on the level of accuracy required for measuring and using decimal and fraction notation when precise measurements are necessary. To demonstrate their understanding of angles as a measure of rotation, the learners will be able to measure and construct angles.
Learning continuums
Mathematics scope and sequence 11
Lear
nin
g c
on
tin
uu
m fo
r m
easu
rem
ent
Ph
ase
1P
has
e 2
Ph
ase
3P
has
e 4
Co
nce
ptu
al u
nd
erst
and
ing
sM
easu
rem
ent i
nvol
ves
com
par
ing
obje
cts
and
even
ts.
Ob
ject
s ha
ve a
ttri
but
es th
at c
an b
e m
easu
red
usin
g no
n-s
tand
ard
unit
s.
Even
ts c
an b
e or
der
ed a
nd s
eque
nced
.
Co
nce
ptu
al u
nd
erst
and
ing
sSt
and
ard
unit
s al
low
us
to h
ave
a co
mm
on
lang
uag
e to
iden
tify
, com
par
e, o
rder
and
se
quen
ce o
bje
cts
and
even
ts.
We
use
tool
s to
mea
sure
the
attr
ibut
es o
f ob
ject
s an
d ev
ents
.
Esti
mat
ion
allo
ws
us to
mea
sure
wit
h d
iffe
rent
leve
ls o
f acc
urac
y.
Co
nce
ptu
al u
nd
erst
and
ing
sO
bje
cts
and
even
ts h
ave
attr
ibut
es th
at
can
be
mea
sure
d us
ing
app
rop
riat
e to
ols.
Rela
tion
ship
s ex
ist b
etw
een
stan
dar
d un
its
that
mea
sure
the
sam
e at
trib
utes
.
Co
nce
ptu
al u
nd
erst
and
ing
sA
ccur
acy
of m
easu
rem
ents
dep
ends
on
the
situ
atio
n an
d th
e p
reci
sion
of t
he to
ol.
Con
vers
ion
of u
nits
and
mea
sure
men
ts
allo
ws
us to
mak
e se
nse
of th
e w
orld
we
live
in.
A ra
nge
of p
roce
dure
s ex
ists
to m
easu
re
dif
fere
nt a
ttri
but
es o
f ob
ject
s an
d ev
ents
.
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
that
att
rib
utes
of r
eal
•ob
ject
s ca
n b
e co
mp
ared
and
d
escr
ibed
, for
exa
mp
le, l
ong
er,
shor
ter,
heav
ier,
emp
ty, f
ull,
hott
er,
cold
er
und
erst
and
that
eve
nts
in d
aily
•
rout
ines
can
be
des
crib
ed a
nd
sequ
ence
d, f
or e
xam
ple
, bef
ore,
aft
er,
bed
tim
e, s
tory
tim
e, to
day
, tom
orro
w.
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
the
use
of s
tand
ard
unit
s •
to m
easu
re, f
or e
xam
ple
, len
gth
, mas
s,
mon
ey, t
ime,
tem
per
atur
e
und
erst
and
that
tool
s ca
n b
e us
ed to
•
mea
sure
und
erst
and
that
cal
end
ars
can
be
•us
ed to
det
erm
ine
the
dat
e, a
nd to
id
enti
fy a
nd s
eque
nce
day
s of
the
wee
k an
d m
onth
s of
the
year
und
erst
and
that
tim
e is
mea
sure
d •
usin
g un
iver
sal u
nits
of m
easu
re, f
or
exam
ple
, yea
rs, m
onth
s, d
ays,
hou
rs,
min
utes
and
sec
onds
.
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
the
use
of s
tand
ard
•un
its
to m
easu
re p
erim
eter
, are
a an
d vo
lum
e
und
erst
and
that
mea
sure
s ca
n fa
ll •
bet
wee
n nu
mb
ers
on a
mea
sure
men
t sc
ale,
for e
xam
ple
, 3½
kg,
bet
wee
n 4
cm a
nd 5
cm
und
erst
and
rela
tion
ship
s b
etw
een
•un
its,
for e
xam
ple
, met
res,
ce
ntim
etre
s an
d m
illim
etre
s
und
erst
and
an a
ngle
as
a m
easu
re o
f •
rota
tion
.
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
pro
cedu
res
for f
ind
ing
•ar
ea, p
erim
eter
and
vol
ume
und
erst
and
the
rela
tion
ship
s b
etw
een
•ar
ea a
nd p
erim
eter
, bet
wee
n ar
ea a
nd
volu
me,
and
bet
wee
n vo
lum
e an
d ca
pac
ity
und
erst
and
unit
con
vers
ions
wit
hin
•m
easu
rem
ent s
yste
ms
(met
ric
or
cust
omar
y).
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
iden
tify
, com
par
e an
d d
escr
ibe
•at
trib
utes
of r
eal o
bje
cts,
for e
xam
ple
, lo
nger
, sho
rter
, hea
vier
, em
pty
, ful
l, ho
tter
, col
der
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
esti
mat
e an
d m
easu
re o
bje
cts
usin
g •
stan
dar
d un
its
of m
easu
rem
ent:
leng
th, m
ass,
cap
acit
y, m
oney
and
te
mp
erat
ure
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
esti
mat
e an
d m
easu
re u
sing
sta
ndar
d •
unit
s of
mea
sure
men
t: p
erim
eter
, are
a an
d vo
lum
e
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
dev
elop
and
des
crib
e fo
rmul
as fo
r •
find
ing
per
imet
er, a
rea
and
volu
me
use
dec
imal
and
frac
tion
not
atio
n in
•
mea
sure
men
t, fo
r exa
mp
le, 3
.2 c
m,
1.47
kg,
1½
mile
s
Learning continuums
Mathematics scope and sequence12
com
par
e th
e le
ngth
, mas
s an
d •
cap
acit
y of
ob
ject
s us
ing
non
-st
and
ard
unit
s
iden
tify
, des
crib
e an
d se
quen
ce
•ev
ents
in th
eir d
aily
rout
ine,
for
exam
ple
, bef
ore,
aft
er, b
edti
me,
st
oryt
ime,
tod
ay, t
omor
row
.
read
and
wri
te th
e ti
me
to th
e ho
ur,
•ha
lf ho
ur a
nd q
uart
er h
our
esti
mat
e an
d co
mp
are
leng
ths
of ti
me:
•
seco
nd, m
inut
e, h
our,
day
, wee
k an
d m
onth
.
des
crib
e m
easu
res
that
fall
bet
wee
n •
num
ber
s on
a s
cale
read
and
wri
te d
igit
al a
nd a
nalo
gue
•ti
me
on 1
2-ho
ur a
nd 2
4-ho
ur c
lock
s.
read
and
inte
rpre
t sca
les
on a
rang
e of
•
mea
suri
ng in
stru
men
ts
mea
sure
and
con
stru
ct a
ngle
s in
•
deg
rees
usi
ng a
pro
trac
tor
carr
y ou
t sim
ple
uni
t con
vers
ions
•
wit
hin
a sy
stem
of m
easu
rem
ent
(met
ric
or c
usto
mar
y).
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
des
crib
e ob
serv
atio
ns a
bou
t eve
nts
•an
d ob
ject
s in
real
-life
sit
uati
ons
use
non
-sta
ndar
d un
its
of
•m
easu
rem
ent t
o so
lve
pro
ble
ms
in
real
-life
sit
uati
ons
invo
lvin
g le
ngth
, m
ass
and
cap
acit
y.
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
use
stan
dar
d un
its
of m
easu
rem
ent t
o •
solv
e p
rob
lem
s in
real
-life
sit
uati
ons
invo
lvin
g le
ngth
, mas
s, c
apac
ity,
m
oney
and
tem
per
atur
e
use
mea
sure
s of
tim
e to
ass
ist w
ith
•p
rob
lem
sol
ving
in re
al-li
fe s
itua
tion
s.
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
use
stan
dar
d un
its
of m
easu
rem
ent t
o •
solv
e p
rob
lem
s in
real
-life
sit
uati
ons
invo
lvin
g p
erim
eter
, are
a an
d vo
lum
e
sele
ct a
pp
rop
riat
e to
ols
and
unit
s of
•
mea
sure
men
t
use
tim
elin
es in
uni
ts o
f inq
uiry
and
•
othe
r rea
l-lif
e si
tuat
ions
.
Whe
n •
app
lyin
g w
ith
un
der
stan
din
g
lear
ners
:
sele
ct a
nd u
se a
pp
rop
riat
e un
its
•of
mea
sure
men
t and
tool
s to
sol
ve
pro
ble
ms
in re
al-li
fe s
itua
tion
s
det
erm
ine
and
just
ify
the
leve
l of
•ac
cura
cy re
quire
d to
sol
ve re
al-li
fe
pro
ble
ms
invo
lvin
g m
easu
rem
ent
use
dec
imal
and
frac
tion
al n
otat
ion
•in
mea
sure
men
t, fo
r exa
mp
le, 3
.2 c
m,
1.47
kg,
1½
mile
s
use
tim
etab
les
and
sche
dule
s (1
2-•
hour
and
24-
hour
clo
cks)
in re
al-li
fe
situ
atio
ns
det
erm
ine
tim
es w
orld
wid
e.•
Learning continuums
Mathematics scope and sequence 13
No
tes
Lear
ners
nee
d m
any
opp
ortu
niti
es to
ex
per
ienc
e an
d qu
anti
fy m
easu
rem
ent
in a
dire
ct k
ines
thet
ic m
anne
r. Th
ey w
ill
dev
elop
und
erst
and
ing
of m
easu
rem
ent
by
usin
g m
anip
ulat
ives
and
mat
eria
ls
from
thei
r im
med
iate
env
ironm
ent,
for
exam
ple
, con
tain
ers
of d
iffe
rent
siz
es,
sand
, wat
er, b
eads
, cor
ks a
nd b
eans
.
No
tes
Usi
ng m
ater
ials
from
thei
r im
med
iate
en
viro
nmen
t, le
arne
rs c
an in
vest
igat
e ho
w u
nits
are
use
d fo
r mea
sure
men
t an
d ho
w m
easu
rem
ents
var
y d
epen
din
g on
the
unit
that
is u
sed
. Lea
rner
s w
ill
refin
e th
eir e
stim
atio
n an
d m
easu
rem
ent
skill
s b
y b
asin
g es
tim
atio
ns o
n p
rior
kn
owle
dg
e, m
easu
ring
the
obje
ct a
nd
com
par
ing
actu
al m
easu
rem
ents
wit
h th
eir e
stim
atio
ns.
No
tes
In o
rder
to u
se m
easu
rem
ent m
ore
auth
enti
cally
, lea
rner
s sh
ould
hav
e th
e op
por
tuni
ty to
mea
sure
real
ob
ject
s in
re
al s
itua
tion
s. T
he u
nits
of i
nqui
ry c
an
ofte
n p
rovi
de
thes
e re
alis
tic
cont
exts
.
A w
ide
rang
e of
mea
suri
ng to
ols
shou
ld
be
avai
lab
le to
the
stud
ents
, for
exa
mp
le,
rule
rs, t
rund
le w
heel
s, ta
pe
mea
sure
s,
bat
hroo
m s
cale
s, k
itche
n sc
ales
, ti
mer
s, a
nalo
gue
cloc
ks, d
igit
al c
lock
s,
stop
wat
ches
and
cal
end
ars.
The
re a
re a
n in
crea
sing
num
ber
of c
omp
uter
and
web
-b
ased
ap
plic
atio
ns a
vaila
ble
for s
tud
ents
to
use
in a
uthe
ntic
con
text
s.
Plea
se n
ote
that
out
com
es re
lati
ng to
an
gles
als
o ap
pea
r in
the
shap
e an
d sp
ace
stra
nd.
No
tes
Lear
ners
gen
eral
ize
thei
r mea
suri
ng
exp
erie
nces
as
they
dev
ise
pro
cedu
res
and
form
ulas
for w
orki
ng o
ut p
erim
eter
, ar
ea a
nd v
olum
e.
Whi
le th
e em
pha
sis
for u
nder
stan
din
g is
on
mea
sure
men
t sys
tem
s co
mm
only
use
d in
the
lear
ner’s
wor
ld, i
t is
wor
thw
hile
b
eing
aw
are
of th
e ex
iste
nce
of o
ther
sy
stem
s an
d ho
w c
onve
rsio
ns b
etw
een
syst
ems
help
us
to m
ake
sens
e of
them
.
Learning continuums
Mathematics scope and sequence14
Shape and spaceThe regions, paths and boundaries of natural space can be described by shape. An understanding of the interrelationships of shape allows us to interpret, understand and appreciate our two-dimensional (2D) and three-dimensional (3D) world.
Overall expectationsPhase 1Learners will understand that shapes have characteristics that can be described and compared. They will understand and use common language to describe paths, regions and boundaries of their immediate environment.
Phase 2Learners will continue to work with 2D and 3D shapes, developing the understanding that shapes are classified and named according to their properties. They will understand that examples of symmetry and transformations can be found in their immediate environment. Learners will interpret, create and use simple directions and specific vocabulary to describe paths, regions, positions and boundaries of their immediate environment.
Phase 3Learners will sort, describe and model regular and irregular polygons, developing an understanding of their properties. They will be able to describe and model congruency and similarity in 2D shapes. Learners will continue to develop their understanding of symmetry, in particular reflective and rotational symmetry. They will understand how geometric shapes and associated vocabulary are useful for representing and describing objects and events in real-world situations.
Phase 4Learners will understand the properties of regular and irregular polyhedra. They will understand the properties of 2D shapes and understand that 2D representations of 3D objects can be used to visualize and solve problems in the real world, for example, through the use of drawing and modelling. Learners will develop their understanding of the use of scale (ratio) to enlarge and reduce shapes. They will apply the language and notation of bearing to describe direction and position.
Learning continuums
Mathematics scope and sequence 15
Lear
nin
g c
on
tin
uu
m fo
r sh
ape
and
sp
ace
Ph
ase
1P
has
e 2
Ph
ase
3P
has
e 4
Co
nce
ptu
al u
nd
erst
and
ing
sSh
apes
can
be
des
crib
ed a
nd o
rgan
ized
ac
cord
ing
to th
eir p
rop
erti
es.
Ob
ject
s in
our
imm
edia
te e
nviro
nmen
t ha
ve a
pos
itio
n in
sp
ace
that
can
be
des
crib
ed a
ccor
din
g to
a p
oint
of
refe
renc
e.
Co
nce
ptu
al u
nd
erst
and
ing
sSh
apes
are
cla
ssifi
ed a
nd n
amed
ac
cord
ing
to th
eir p
rop
erti
es.
Som
e sh
apes
are
mad
e up
of p
arts
that
re
pea
t in
som
e w
ay.
Spec
ific
voca
bul
ary
can
be
used
to
des
crib
e an
ob
ject
’s p
osit
ion
in s
pac
e.
Co
nce
ptu
al u
nd
erst
and
ing
sC
hang
ing
the
pos
itio
n of
a s
hap
e d
oes
not a
lter
its
pro
per
ties
.
Shap
es c
an b
e tr
ansf
orm
ed in
dif
fere
nt
way
s.
Geo
met
ric
shap
es a
nd v
ocab
ular
y ar
e us
eful
for r
epre
sent
ing
and
des
crib
ing
obje
cts
and
even
ts in
real
-wor
ld
situ
atio
ns.
Co
nce
ptu
al u
nd
erst
and
ing
sM
anip
ulat
ion
of s
hap
e an
d sp
ace
take
s p
lace
for a
par
ticu
lar p
urp
ose.
Con
solid
atin
g w
hat w
e kn
ow o
f g
eom
etri
c co
ncep
ts a
llow
s us
to m
ake
sens
e of
and
inte
ract
wit
h ou
r wor
ld.
Geo
met
ric
tool
s an
d m
etho
ds c
an b
e us
ed to
sol
ve p
rob
lem
s re
lati
ng to
sha
pe
and
spac
e.
Lear
nin
g o
utc
om
es
Whe
n co
nst
ruct
ing
mea
nin
g le
arne
rs:
und
erst
and
that
2D
and
3D
sha
pes
•
have
cha
ract
eris
tics
that
can
be
des
crib
ed a
nd c
omp
ared
und
erst
and
that
com
mon
lang
uag
e •
can
be
used
to d
escr
ibe
pos
itio
n an
d d
irect
ion,
for e
xam
ple
, ins
ide,
out
sid
e,
abov
e, b
elow
, nex
t to,
beh
ind
, in
fron
t of
, up,
dow
n.
Lear
nin
g o
utc
om
es
Whe
n co
nst
ruct
ing
mea
nin
g le
arne
rs:
und
erst
and
that
ther
e ar
e •
rela
tion
ship
s am
ong
and
bet
wee
n 2D
an
d 3D
sha
pes
und
erst
and
that
2D
and
3D
sha
pes
•
can
be
crea
ted
by
put
ting
tog
ethe
r an
d/o
r tak
ing
apar
t oth
er s
hap
es
und
erst
and
that
exa
mp
les
of
•sy
mm
etry
and
tran
sfor
mat
ions
can
be
foun
d in
thei
r im
med
iate
env
ironm
ent
und
erst
and
that
geo
met
ric
shap
es
•ar
e us
eful
for r
epre
sent
ing
real
-wor
ld
situ
atio
ns
und
erst
and
that
dire
ctio
ns c
an b
e •
used
to d
escr
ibe
pat
hway
s, re
gion
s,
pos
itio
ns a
nd b
ound
arie
s of
thei
r im
med
iate
env
ironm
ent.
Lear
nin
g o
utc
om
es
Whe
n co
nst
ruct
ing
mea
nin
g le
arne
rs:
und
erst
and
the
com
mon
lang
uag
e •
used
to d
escr
ibe
shap
es
und
erst
and
the
pro
per
ties
of r
egul
ar
•an
d ir
regu
lar p
olyg
ons
und
erst
and
cong
ruen
t or s
imila
r •
shap
es
und
erst
and
that
line
s an
d ax
es o
f •
refl
ecti
ve a
nd ro
tati
onal
sym
met
ry
assi
st w
ith
the
cons
truc
tion
of s
hap
es
und
erst
and
an a
ngle
as
a m
easu
re o
f •
rota
tion
und
erst
and
that
dire
ctio
ns fo
r •
loca
tion
can
be
rep
rese
nted
by
coor
din
ates
on
a gr
id
und
erst
and
that
vis
ualiz
atio
n of
sha
pe
•an
d sp
ace
is a
str
ateg
y fo
r sol
ving
p
rob
lem
s.
Lear
nin
g o
utc
om
es
Whe
n co
nst
ruct
ing
mea
nin
g le
arne
rs:
und
erst
and
the
com
mon
lang
uag
e •
used
to d
escr
ibe
shap
es
und
erst
and
the
pro
per
ties
of r
egul
ar
•an
d ir
regu
lar p
olyh
edra
und
erst
and
the
pro
per
ties
of c
ircle
s•
und
erst
and
how
sca
le (r
atio
s) is
use
d •
to e
nlar
ge
and
redu
ce s
hap
es
und
erst
and
syst
ems
for d
escr
ibin
g •
pos
itio
n an
d d
irect
ion
und
erst
and
that
2D
rep
rese
ntat
ions
•
of 3
D o
bje
cts
can
be
used
to v
isua
lize
and
solv
e p
rob
lem
s
und
erst
and
that
geo
met
ric
idea
s •
and
rela
tion
ship
s ca
n b
e us
ed to
so
lve
pro
ble
ms
in o
ther
are
as o
f m
athe
mat
ics
and
in re
al li
fe.
Learning continuums
Mathematics scope and sequence16
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
sort
, des
crib
e an
d co
mp
are
3D s
hap
es
•
des
crib
e p
osit
ion
and
dire
ctio
n, fo
r •
exam
ple
, ins
ide,
out
sid
e, a
bov
e,
bel
ow, n
ext t
o, b
ehin
d, i
n fr
ont o
f, up
, d
own.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
sort
, des
crib
e an
d la
bel
2D
and
3D
•
shap
es
anal
yse
and
des
crib
e th
e re
lati
onsh
ips
•b
etw
een
2D a
nd 3
D s
hap
es
crea
te a
nd d
escr
ibe
sym
met
rica
l and
•
tess
ella
ting
pat
tern
s
iden
tify
line
s of
refl
ecti
ve s
ymm
etry
•
rep
rese
nt id
eas
abou
t the
real
wor
ld
•us
ing
geo
met
ric
voca
bul
ary
and
sym
bol
s, fo
r exa
mp
le, t
hrou
gh o
ral
des
crip
tion
, dra
win
g, m
odel
ling,
la
bel
ling
inte
rpre
t and
cre
ate
sim
ple
dire
ctio
ns,
•d
escr
ibin
g p
aths
, reg
ions
, pos
itio
ns
and
bou
ndar
ies
of th
eir i
mm
edia
te
envi
ronm
ent.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
sort
, des
crib
e an
d m
odel
regu
lar a
nd
•ir
regu
lar p
olyg
ons
des
crib
e an
d m
odel
con
grue
ncy
and
•si
mila
rity
in 2
D s
hap
es
anal
yse
angl
es b
y co
mp
arin
g an
d •
des
crib
ing
rota
tion
s: w
hole
turn
; hal
f tu
rn; q
uart
er tu
rn; n
orth
, sou
th, e
ast
and
wes
t on
a co
mp
ass
loca
te fe
atur
es o
n a
grid
usi
ng
•co
ord
inat
es
des
crib
e an
d/o
r rep
rese
nt m
enta
l •
imag
es o
f ob
ject
s, p
atte
rns,
and
p
aths
.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
anal
yse,
des
crib
e, c
lass
ify
and
visu
aliz
e •
2D (i
nclu
din
g ci
rcle
s, tr
iang
les
and
quad
rila
tera
ls) a
nd 3
D s
hap
es, u
sing
g
eom
etri
c vo
cab
ular
y
des
crib
e lin
es a
nd a
ngle
s us
ing
•g
eom
etri
c vo
cab
ular
y
iden
tify
and
use
sca
le (r
atio
s) to
•
enla
rge
and
redu
ce s
hap
es
iden
tify
and
use
the
lang
uag
e an
d •
nota
tion
of b
eari
ng to
des
crib
e d
irect
ion
and
pos
itio
n
crea
te a
nd m
odel
how
a 2
D n
et
•co
nver
ts in
to a
3D
sha
pe
and
vice
ve
rsa
exp
lore
the
use
of g
eom
etri
c id
eas
•an
d re
lati
onsh
ips
to s
olve
pro
ble
ms
in
othe
r are
as o
f mat
hem
atic
s.
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
exp
lore
and
des
crib
e th
e p
aths
, •
regi
ons
and
bou
ndar
ies
of th
eir
imm
edia
te e
nviro
nmen
t (in
sid
e,
outs
ide,
ab
ove,
bel
ow) a
nd th
eir
pos
itio
n (n
ext t
o, b
ehin
d, i
n fr
ont o
f, up
, dow
n).
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
anal
yse
and
use
wha
t the
y kn
ow
•ab
out 3
D s
hap
es to
des
crib
e an
d w
ork
wit
h 2D
sha
pes
reco
gniz
e an
d ex
pla
in s
imp
le
•sy
mm
etri
cal d
esig
ns in
the
envi
ronm
ent
app
ly k
now
led
ge
of s
ymm
etry
to
•p
rob
lem
-sol
ving
sit
uati
ons
inte
rpre
t and
use
sim
ple
dire
ctio
ns,
•d
escr
ibin
g p
aths
, reg
ions
, pos
itio
ns
and
bou
ndar
ies
of th
eir i
mm
edia
te
envi
ronm
ent.
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
anal
yse
and
des
crib
e 2D
and
3D
•
shap
es, i
nclu
din
g re
gula
r and
ir
regu
lar p
olyg
ons,
usi
ng g
eom
etri
cal
voca
bul
ary
iden
tify
, des
crib
e an
d m
odel
•
cong
ruen
cy a
nd s
imila
rity
in 2
D
shap
es
reco
gniz
e an
d ex
pla
in s
ymm
etri
cal
•p
atte
rns,
incl
udin
g te
ssel
lati
on, i
n th
e en
viro
nmen
t
app
ly k
now
led
ge
of tr
ansf
orm
atio
ns
•to
pro
ble
m-s
olvi
ng s
itua
tion
s.
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
use
geo
met
ric
voca
bul
ary
whe
n •
des
crib
ing
shap
e an
d sp
ace
in
mat
hem
atic
al s
itua
tion
s an
d b
eyon
d
use
scal
e (r
atio
s) to
enl
arg
e an
d •
redu
ce s
hap
es
app
ly th
e la
ngua
ge
and
nota
tion
of
•b
eari
ng to
des
crib
e d
irect
ion
and
pos
itio
n
use
2D re
pre
sent
atio
ns o
f 3D
ob
ject
s •
to v
isua
lize
and
solv
e p
rob
lem
s, fo
r ex
amp
le u
sing
dra
win
gs o
r mod
els.
Learning continuums
Mathematics scope and sequence 17
No
tes
Lear
ners
nee
d m
any
opp
ortu
niti
es to
ex
per
ienc
e sh
ape
and
spac
e in
a d
irect
ki
nest
heti
c m
anne
r, fo
r exa
mp
le, t
hrou
gh
pla
y, c
onst
ruct
ion
and
mov
emen
t.
The
man
ipul
ativ
es th
at th
ey in
tera
ct w
ith
shou
ld in
clud
e a
rang
e of
3D
sha
pes
, in
par
ticu
lar t
he re
al-li
fe o
bje
cts
wit
h w
hich
ch
ildre
n ar
e fa
mili
ar. 2
D s
hap
es (p
lane
sh
apes
) are
a m
ore
abst
ract
con
cep
t but
ca
n b
e un
der
stoo
d as
face
s of
3D
sha
pes
.
No
tes
Lear
ners
nee
d to
und
erst
and
the
pro
per
ties
of 2
D a
nd 3
D s
hap
es
bef
ore
the
mat
hem
atic
al v
ocab
ular
y as
soci
ated
wit
h sh
apes
mak
es s
ense
to
them
. Thr
ough
cre
atin
g an
d m
anip
ulat
ing
shap
es, l
earn
ers
alig
n th
eir
natu
ral v
ocab
ular
y w
ith
mor
e fo
rmal
m
athe
mat
ical
voc
abul
ary
and
beg
in to
ap
pre
ciat
e th
e ne
ed fo
r thi
s p
reci
sion
.
No
tes
Com
put
er a
nd w
eb-b
ased
ap
plic
atio
ns
can
be
used
to e
xplo
re s
hap
e an
d sp
ace
conc
epts
suc
h as
sym
met
ry, a
ngle
s an
d co
ord
inat
es.
The
unit
s of
inqu
iry
can
pro
vid
e au
then
tic
cont
exts
for d
evel
opin
g un
der
stan
din
g of
con
cep
ts re
lati
ng to
loca
tion
and
d
irect
ions
.
No
tes
Tool
s su
ch a
s co
mp
asse
s an
d p
rotr
acto
rs
are
com
mon
ly u
sed
to s
olve
pro
ble
ms
in
real
-life
sit
uati
ons.
How
ever
, car
e sh
ould
b
e ta
ken
to e
nsur
e th
at s
tud
ents
hav
e a
stro
ng u
nder
stan
din
g of
the
conc
epts
em
bed
ded
in th
e p
rob
lem
to e
nsur
e m
eani
ngfu
l eng
agem
ent w
ith
the
tool
s an
d fu
ll un
der
stan
din
g of
the
solu
tion
.
Learning continuums
Mathematics scope and sequence18
Pattern and functionTo identify pattern is to begin to understand how mathematics applies to the world in which we live. The repetitive features of patterns can be identified and described as generalized rules called “functions”. This builds a foundation for the later study of algebra.
Overall expectations Phase 1Learners will understand that patterns and sequences occur in everyday situations. They will be able to identify, describe, extend and create patterns in various ways.
Phase 2Learners will understand that whole numbers exhibit patterns and relationships that can be observed and described, and that the patterns can be represented using numbers and other symbols. As a result, learners will understand the inverse relationship between addition and subtraction, and the associative and commutative properties of addition. They will be able to use their understanding of pattern to represent and make sense of real-life situations and, where appropriate, to solve problems involving addition and subtraction.
Phase 3Learners will analyse patterns and identify rules for patterns, developing the understanding that functions describe the relationship or rules that uniquely associate members of one set with members of another set. They will understand the inverse relationship between multiplication and division, and the associative and commutative properties of multiplication. They will be able to use their understanding of pattern and function to represent and make sense of real-life situations and, where appropriate, to solve problems involving the four operations.
Phase 4Learners will understand that patterns can be represented, analysed and generalized using algebraic expressions, equations or functions. They will use words, tables, graphs and, where possible, symbolic rules to analyse and represent patterns. They will develop an understanding of exponential notation as a way to express repeated products, and of the inverse relationship that exists between exponents and roots. The students will continue to use their understanding of pattern and function to represent and make sense of real-life situations and to solve problems involving the four operations.
Learning continuums
Mathematics scope and sequence 19
Lear
nin
g c
on
tin
uu
m fo
r p
atte
rn a
nd
fun
ctio
n
Ph
ase
1P
has
e 2
Ph
ase
3P
has
e 4
Co
nce
ptu
al u
nd
erst
and
ing
s
Patt
erns
and
seq
uenc
es o
ccur
in e
very
day
situ
atio
ns.
Patt
erns
rep
eat a
nd g
row
.
Co
nce
ptu
al u
nd
erst
and
ing
s
Who
le n
umb
ers
exhi
bit
pat
tern
s an
d
rela
tion
ship
s th
at c
an b
e ob
serv
ed a
nd
des
crib
ed.
Patt
erns
can
be
rep
rese
nted
usi
ng
num
ber
s an
d ot
her s
ymb
ols.
Co
nce
ptu
al u
nd
erst
and
ing
s
Func
tion
s ar
e re
lati
onsh
ips
or ru
les
that
uniq
uely
ass
ocia
te m
emb
ers
of o
ne s
et
wit
h m
emb
ers
of a
noth
er s
et.
By a
naly
sing
pat
tern
s an
d id
enti
fyin
g
rule
s fo
r pat
tern
s it
is p
ossi
ble
to m
ake
pre
dic
tion
s.
Co
nce
ptu
al u
nd
erst
and
ing
s
Patt
erns
can
oft
en b
e g
ener
aliz
ed u
sing
alg
ebra
ic e
xpre
ssio
ns, e
quat
ions
or
func
tion
s.
Exp
onen
tial
not
atio
n is
a p
ower
ful w
ay
to e
xpre
ss re
pea
ted
pro
duct
s of
the
sam
e
num
ber
.
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
that
pat
tern
s ca
n b
e •
foun
d in
eve
ryd
ay s
itua
tion
s, fo
r ex
amp
le, s
ound
s, a
ctio
ns, o
bje
cts,
na
ture
.
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
that
pat
tern
s ca
n b
e •
foun
d in
num
ber
s, fo
r exa
mp
le, o
dd
and
even
num
ber
s, s
kip
cou
ntin
g
und
erst
and
the
inve
rse
rela
tion
ship
•
bet
wee
n ad
dit
ion
and
sub
trac
tion
und
erst
and
the
asso
ciat
ive
and
•co
mm
utat
ive
pro
per
ties
of a
dd
itio
n.
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
that
pat
tern
s ca
n b
e •
anal
ysed
and
rule
s id
enti
fied
und
erst
and
that
mul
tip
licat
ion
is
•re
pea
ted
add
itio
n an
d th
at d
ivis
ion
is
rep
eate
d su
btr
acti
on
und
erst
and
the
inve
rse
rela
tion
ship
•
bet
wee
n m
ulti
plic
atio
n an
d d
ivis
ion
und
erst
and
the
asso
ciat
ive
•an
d co
mm
utat
ive
pro
per
ties
of
mul
tip
licat
ion.
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
that
pat
tern
s ca
n b
e •
gen
eral
ized
by
a ru
le
und
erst
and
exp
onen
ts a
s re
pea
ted
•m
ulti
plic
atio
n
und
erst
and
the
inve
rse
rela
tion
ship
•
bet
wee
n ex
pon
ents
and
root
s
und
erst
and
that
pat
tern
s ca
n b
e •
rep
rese
nted
, ana
lyse
d an
d g
ener
aliz
ed
usin
g ta
ble
s, g
rap
hs, w
ords
, and
, w
hen
pos
sib
le, s
ymb
olic
rule
s.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
des
crib
e p
atte
rns
in v
ario
us w
ays,
•
for e
xam
ple
, usi
ng w
ords
, dra
win
gs,
sym
bol
s, m
ater
ials
, act
ions
, num
ber
s.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
rep
rese
nt p
atte
rns
in a
var
iety
of w
ays,
•
for e
xam
ple
, usi
ng w
ords
, dra
win
gs,
sym
bol
s, m
ater
ials
, act
ions
, num
ber
s
des
crib
e nu
mb
er p
atte
rns,
for
•ex
amp
le, o
dd
and
even
num
ber
s, s
kip
co
unti
ng.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
des
crib
e th
e ru
le fo
r a p
atte
rn in
a
•va
riet
y of
way
s
rep
rese
nt ru
les
for p
atte
rns
usin
g •
wor
ds, s
ymb
ols
and
tab
les
iden
tify
a s
eque
nce
of o
per
atio
ns
•re
lati
ng o
ne s
et o
f num
ber
s to
ano
ther
se
t.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
rep
rese
nt th
e ru
le o
f a p
atte
rn b
y •
usin
g a
func
tion
anal
yse
pat
tern
and
func
tion
usi
ng
•w
ords
, tab
les
and
grap
hs, a
nd, w
hen
pos
sib
le, s
ymb
olic
rule
s.
Learning continuums
Mathematics scope and sequence20
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
exte
nd a
nd c
reat
e p
atte
rns.
•
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
exte
nd a
nd c
reat
e p
atte
rns
in
•nu
mb
ers,
for e
xam
ple
, od
d an
d ev
en
num
ber
s, s
kip
cou
ntin
g
use
num
ber
pat
tern
s to
rep
rese
nt a
nd
•un
der
stan
d re
al-li
fe s
itua
tion
s
use
the
pro
per
ties
and
rela
tion
ship
s •
of a
dd
itio
n an
d su
btr
acti
on to
sol
ve
pro
ble
ms.
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
sele
ct a
pp
rop
riat
e m
etho
ds fo
r •
rep
rese
ntin
g p
atte
rns,
for e
xam
ple
us
ing
wor
ds, s
ymb
ols
and
tab
les
use
num
ber
pat
tern
s to
mak
e •
pre
dic
tion
s an
d so
lve
pro
ble
ms
use
the
pro
per
ties
and
rela
tion
ship
s of
•
the
four
op
erat
ions
to s
olve
pro
ble
ms.
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
le
arne
rs:
sele
ct a
pp
rop
riat
e m
etho
ds to
ana
lyse
•
pat
tern
s an
d id
enti
fy ru
les
use
func
tion
s to
sol
ve p
rob
lem
s.•
No
tes
The
wor
ld is
fille
d w
ith
pat
tern
and
ther
e w
ill b
e m
any
opp
ortu
niti
es fo
r lea
rner
s to
mak
e th
is c
onne
ctio
n ac
ross
the
curr
icul
um.
A ra
nge
of m
anip
ulat
ives
can
be
used
to
exp
lore
pat
tern
s in
clud
ing
pat
tern
blo
cks,
at
trib
ute
blo
cks,
col
our t
iles,
cal
cula
tors
, nu
mb
er c
hart
s, b
eans
and
but
tons
.
No
tes
Stud
ents
will
ap
ply
thei
r und
erst
and
ing
of p
atte
rn to
the
num
ber
s th
ey a
lread
y kn
ow. T
he p
atte
rns
they
find
will
hel
p to
d
eep
en th
eir u
nder
stan
din
g of
a ra
nge
of
num
ber
con
cep
ts.
Four
-fun
ctio
n ca
lcul
ator
s ca
n b
e us
ed to
ex
plo
re n
umb
er p
atte
rns.
No
tes
Patt
erns
are
cen
tral
to th
e un
der
stan
din
g of
all
conc
epts
in m
athe
mat
ics.
The
y ar
e th
e b
asis
of h
ow o
ur n
umb
er s
yste
m is
or
gani
zed
. Sea
rchi
ng fo
r, an
d id
enti
fyin
g,
pat
tern
s he
lps
us to
see
rela
tion
ship
s,
mak
e g
ener
aliz
atio
ns, a
nd is
a p
ower
ful
stra
teg
y fo
r pro
ble
m s
olvi
ng. F
unct
ions
d
evel
op fr
om th
e st
udy
of p
atte
rns
and
mak
e it
pos
sib
le to
pre
dic
t in
mat
hem
atic
s p
rob
lem
s.
No
tes
Alg
ebra
is a
mat
hem
atic
al la
ngua
ge
usin
g nu
mb
ers
and
sym
bol
s to
exp
ress
re
lati
onsh
ips.
Whe
n th
e sa
me
rela
tion
ship
w
orks
wit
h an
y nu
mb
er, a
lgeb
ra u
ses
lett
ers
to re
pre
sent
the
gen
eral
izat
ion.
Le
tter
s ca
n b
e us
ed to
rep
rese
nt th
e qu
anti
ty.
Learning continuums
Mathematics scope and sequence 21
NumberOur number system is a language for describing quantities and the relationships between quantities. For example, the value attributed to a digit depends on its place within a base system.
Numbers are used to interpret information, make decisions and solve problems. For example, the operations of addition, subtraction, multiplication and division are related to one another and are used to process information in order to solve problems. The degree of precision needed in calculating depends on how the result will be used.
Overall expectationsPhase 1Learners will understand that numbers are used for many different purposes in the real world. They will develop an understanding of one-to-one correspondence and conservation of number, and be able to count and use number words and numerals to represent quantities.
Phase 2Learners will develop their understanding of the base 10 place value system and will model, read, write, estimate, compare and order numbers to hundreds or beyond. They will have automatic recall of addition and subtraction facts and be able to model addition and subtraction of whole numbers using the appropriate mathematical language to describe their mental and written strategies. Learners will have an understanding of fractions as representations of whole-part relationships and will be able to model fractions and use fraction names in real-life situations.
Phase 3Learners will develop the understanding that fractions and decimals are ways of representing whole-part relationships and will demonstrate this understanding by modelling equivalent fractions and decimal fractions to hundredths or beyond. They will be able to model, read, write, compare and order fractions, and use them in real-life situations. Learners will have automatic recall of addition, subtraction, multiplication and division facts. They will select, use and describe a range of strategies to solve problems involving addition, subtraction, multiplication and division, using estimation strategies to check the reasonableness of their answers.
Phase 4Learners will understand that the base 10 place value system extends infinitely in two directions and will be able to model, compare, read, write and order numbers to millions or beyond, as well as model integers. They will develop an understanding of ratios. They will understand that fractions, decimals and percentages are ways of representing whole-part relationships and will work towards modelling, comparing, reading, writing, ordering and converting fractions, decimals and percentages. They will use mental and written strategies to solve problems involving whole numbers, fractions and decimals in real-life situations, using a range of strategies to evaluate reasonableness of answers.
Learning continuums
Mathematics scope and sequence22
Lear
nin
g c
on
tin
uu
m fo
r n
um
ber
Ph
ase
1P
has
e 2
Ph
ase
3P
has
e 4
Co
nce
ptu
al u
nd
erst
and
ing
sN
umb
ers
are
a na
min
g sy
stem
.
Num
ber
s ca
n b
e us
ed in
man
y w
ays
for
dif
fere
nt p
urp
oses
in th
e re
al w
orld
.
Num
ber
s ar
e co
nnec
ted
to e
ach
othe
r th
roug
h a
vari
ety
of re
lati
onsh
ips.
Mak
ing
conn
ecti
ons
bet
wee
n ou
r ex
per
ienc
es w
ith
num
ber
can
hel
p u
s to
d
evel
op n
umb
er s
ense
.
Co
nce
ptu
al u
nd
erst
and
ing
sTh
e b
ase
10 p
lace
val
ue s
yste
m is
use
d to
rep
rese
nt n
umb
ers
and
num
ber
re
lati
onsh
ips.
Frac
tion
s ar
e w
ays
of re
pre
sent
ing
who
le-
par
t rel
atio
nshi
ps.
The
oper
atio
ns o
f ad
dit
ion,
sub
trac
tion
, m
ulti
plic
atio
n an
d d
ivis
ion
are
rela
ted
to e
ach
othe
r and
are
use
d to
pro
cess
in
form
atio
n to
sol
ve p
rob
lem
s.
Num
ber
op
erat
ions
can
be
mod
elle
d in
a
vari
ety
of w
ays.
Ther
e ar
e m
any
men
tal m
etho
ds th
at c
an
be
app
lied
for e
xact
and
ap
pro
xim
ate
com
put
atio
ns.
Co
nce
ptu
al u
nd
erst
and
ing
sTh
e b
ase
10 p
lace
val
ue s
yste
m c
an b
e ex
tend
ed to
rep
rese
nt m
agni
tud
e.
Frac
tion
s an
d d
ecim
als
are
way
s of
re
pre
sent
ing
who
le-p
art r
elat
ions
hip
s.
The
oper
atio
ns o
f ad
dit
ion,
sub
trac
tion
, m
ulti
plic
atio
n an
d d
ivis
ion
are
rela
ted
to e
ach
othe
r and
are
use
d to
pro
cess
in
form
atio
n to
sol
ve p
rob
lem
s.
Even
com
ple
x op
erat
ions
can
be
mod
elle
d in
a v
arie
ty o
f way
s, fo
r ex
amp
le, a
n al
gor
ithm
is a
way
to
rep
rese
nt a
n op
erat
ion.
Co
nce
ptu
al u
nd
erst
and
ing
sTh
e b
ase
10 p
lace
val
ue s
yste
m e
xten
ds
infin
itely
in t
wo
dire
ctio
ns.
Frac
tion
s, d
ecim
al fr
acti
ons
and
per
cent
ages
are
way
s of
rep
rese
ntin
g w
hole
-par
t rel
atio
nshi
ps.
For f
ract
iona
l and
dec
imal
com
put
atio
n,
the
idea
s d
evel
oped
for w
hole
-num
ber
co
mp
utat
ion
can
app
ly.
Rati
os a
re a
com
par
ison
of t
wo
num
ber
s or
qua
ntit
ies.
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
und
erst
and
one-
to-o
ne
•co
rres
pon
den
ce
und
erst
and
that
, for
a s
et o
f ob
ject
s,
•th
e nu
mb
er n
ame
of th
e la
st o
bje
ct
coun
ted
des
crib
es th
e qu
anti
ty o
f the
w
hole
set
und
erst
and
that
num
ber
s ca
n •
be
cons
truc
ted
in m
ulti
ple
way
s,
for e
xam
ple
, by
com
bin
ing
and
par
titi
onin
g
und
erst
and
cons
erva
tion
of n
umb
er*
•
und
erst
and
the
rela
tive
mag
nitu
de
of
•w
hole
num
ber
s
reco
gniz
e gr
oup
s of
zer
o to
five
•
obje
cts
wit
hout
cou
ntin
g (s
ubit
izin
g)
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
mod
el n
umb
ers
to h
und
reds
or
•b
eyon
d us
ing
the
bas
e 10
pla
ce v
alue
sy
stem
**
esti
mat
e qu
anti
ties
to 1
00 o
r bey
ond
•
mod
el s
imp
le fr
acti
on re
lati
onsh
ips
•
use
the
lang
uag
e of
ad
dit
ion
and
•su
btr
acti
on, f
or e
xam
ple
, ad
d, t
ake
away
, plu
s, m
inus
, sum
, dif
fere
nce
mod
el a
dd
itio
n an
d su
btr
acti
on o
f •
who
le n
umb
ers
dev
elop
str
ateg
ies
for m
emor
izin
g •
add
itio
n an
d su
btr
acti
on n
umb
er
fact
s
esti
mat
e su
ms
and
dif
fere
nces
•
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
mod
el n
umb
ers
to th
ousa
nds
or
•b
eyon
d us
ing
the
bas
e 10
pla
ce v
alue
sy
stem
mod
el e
quiv
alen
t fra
ctio
ns•
use
the
lang
uag
e of
frac
tion
s, fo
r •
exam
ple
, num
erat
or, d
enom
inat
or
mod
el d
ecim
al fr
acti
ons
to
•hu
ndre
dth
s or
bey
ond
mod
el m
ulti
plic
atio
n an
d d
ivis
ion
of
•w
hole
num
ber
s
use
the
lang
uag
e of
mul
tip
licat
ion
•an
d d
ivis
ion,
for e
xam
ple
, fac
tor,
mul
tip
le, p
rodu
ct, q
uoti
ent,
pri
me
num
ber
s, c
omp
osite
num
ber
Lear
nin
g o
utc
om
esW
hen
con
stru
ctin
g m
ean
ing
lear
ners
:
mod
el n
umb
ers
to m
illio
ns o
r bey
ond
•us
ing
the
bas
e 10
pla
ce v
alue
sys
tem
mod
el ra
tios
•
mod
el in
teg
ers
in a
pp
rop
riat
e •
cont
exts
mod
el e
xpon
ents
and
squ
are
root
s •
mod
el im
pro
per
frac
tion
s an
d m
ixed
•
num
ber
s
sim
plif
y fr
acti
ons
usin
g m
anip
ulat
ives
•
mod
el d
ecim
al fr
acti
ons
to
•th
ousa
ndth
s or
bey
ond
mod
el p
erce
ntag
es•
und
erst
and
the
rela
tion
ship
bet
wee
n •
frac
tion
s, d
ecim
als
and
per
cent
ages
Learning continuums
Mathematics scope and sequence 23
und
erst
and
who
le-p
art r
elat
ions
hip
s •
use
the
lang
uag
e of
mat
hem
atic
s •
to c
omp
are
quan
titi
es, f
or e
xam
ple
, m
ore,
less
, fir
st, s
econ
d.
und
erst
and
situ
atio
ns th
at in
volv
e •
mul
tip
licat
ion
and
div
isio
n
mod
el a
dd
itio
n an
d su
btr
acti
on o
f •
frac
tion
s w
ith
the
sam
e d
enom
inat
or.
mod
el a
dd
itio
n an
d su
btr
acti
on
•of
frac
tion
s w
ith
rela
ted
den
omin
ator
s***
mod
el a
dd
itio
n an
d su
btr
acti
on o
f •
dec
imal
s.
mod
el a
dd
itio
n, s
ubtr
acti
on,
•m
ulti
plic
atio
n an
d d
ivis
ion
of fr
acti
ons
mod
el a
dd
itio
n, s
ubtr
acti
on,
•m
ulti
plic
atio
n an
d d
ivis
ion
of
dec
imal
s.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
conn
ect n
umb
er n
ames
and
•
num
eral
s to
the
quan
titie
s th
ey
rep
rese
nt.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
read
and
wri
te w
hole
num
ber
s up
to
•hu
ndre
ds o
r bey
ond
read
, wri
te, c
omp
are
and
ord
er
•ca
rdin
al a
nd o
rdin
al n
umb
ers
des
crib
e m
enta
l and
wri
tten
•
stra
tegi
es fo
r ad
din
g an
d su
btr
acti
ng
two
-dig
it n
umb
ers.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
read
, wri
te, c
omp
are
and
ord
er w
hole
•
num
ber
s up
to th
ousa
nds
or b
eyon
d
dev
elop
str
ateg
ies
for m
emor
izin
g •
add
itio
n, s
ubtr
acti
on, m
ulti
plic
atio
n an
d d
ivis
ion
num
ber
fact
s
read
, wri
te, c
omp
are
and
ord
er
•fr
acti
ons
read
and
wri
te e
quiv
alen
t fra
ctio
ns•
read
, wri
te, c
omp
are
and
ord
er
•fr
acti
ons
to h
und
red
ths
or b
eyon
d
des
crib
e m
enta
l and
wri
tten
•
stra
tegi
es fo
r mul
tip
licat
ion
and
div
isio
n.
Whe
n tr
ansf
erri
ng
mea
nin
g in
to
sym
bo
ls le
arne
rs:
read
, wri
te, c
omp
are
and
ord
er w
hole
•
num
ber
s up
to m
illio
ns o
r bey
ond
read
and
wri
te ra
tios
•
read
and
wri
te in
teg
ers
in a
pp
rop
riat
e •
cont
exts
read
and
wri
te e
xpon
ents
and
squ
are
•ro
ots
conv
ert i
mp
rop
er fr
acti
ons
to m
ixed
•
num
ber
s an
d vi
ce v
ersa
sim
plif
y fr
acti
ons
in m
enta
l and
•
wri
tten
form
read
, wri
te, c
omp
are
and
ord
er
•d
ecim
al fr
acti
ons
to th
ousa
ndth
s or
b
eyon
d
read
, wri
te, c
omp
are
and
ord
er
•p
erce
ntag
es
conv
ert b
etw
een
frac
tion
s, d
ecim
als
•an
d p
erce
ntag
es.
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
lear
ners
:
coun
t to
det
erm
ine
the
num
ber
of
•ob
ject
s in
a s
et
use
num
ber
wor
ds a
nd n
umer
als
•to
rep
rese
nt q
uant
itie
s in
real
-life
si
tuat
ions
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
lear
ners
:
use
who
le n
umb
ers
up to
hun
dre
ds o
r •
bey
ond
in re
al-li
fe s
itua
tion
s
use
card
inal
and
ord
inal
num
ber
s in
•
real
-life
sit
uati
ons
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
lear
ners
:
use
who
le n
umb
ers
up to
thou
sand
s •
or b
eyon
d in
real
-life
sit
uati
ons
use
fast
reca
ll of
mul
tip
licat
ion
and
•d
ivis
ion
num
ber
fact
s in
real
-life
si
tuat
ions
Whe
n ap
ply
ing
wit
h u
nd
erst
and
ing
lear
ners
:
use
who
le n
umb
ers
up to
mill
ions
or
•b
eyon
d in
real
-life
sit
uati
ons
use
rati
os in
real
-life
sit
uati
ons
•
use
inte
ger
s in
real
-life
sit
uati
ons
•
Learning continuums
Mathematics scope and sequence24
use
the
lang
uag
e of
mat
hem
atic
s •
to c
omp
are
quan
titi
es in
real
-life
si
tuat
ions
, for
exa
mp
le, m
ore,
less
, fir
st, s
econ
d
sub
itiz
e in
real
-life
sit
uati
ons
•
use
sim
ple
frac
tion
nam
es in
real
-life
•
situ
atio
ns.
use
fast
reca
ll of
ad
dit
ion
and
•su
btr
acti
on n
umb
er fa
cts
in re
al-li
fe
situ
atio
ns
use
frac
tion
s in
real
-life
sit
uati
ons
•
use
men
tal a
nd w
ritt
en s
trat
egie
s fo
r •
add
itio
n an
d su
btr
acti
on o
f tw
o-
dig
it n
umb
ers
or b
eyon
d in
real
-life
si
tuat
ions
sele
ct a
n ap
pro
pri
ate
met
hod
for
•so
lvin
g a
pro
ble
m, f
or e
xam
ple
, m
enta
l est
imat
ion,
men
tal o
r wri
tten
st
rate
gies
, or b
y us
ing
a ca
lcul
ator
use
stra
tegi
es to
eva
luat
e th
e •
reas
onab
lene
ss o
f ans
wer
s.
use
dec
imal
frac
tion
s in
real
-life
•
situ
atio
ns
use
men
tal a
nd w
ritt
en s
trat
egie
s fo
r •
mul
tip
licat
ion
and
div
isio
n in
real
-life
si
tuat
ions
sele
ct a
n ef
ficie
nt m
etho
d fo
r •
solv
ing
a p
rob
lem
, for
exa
mp
le,
men
tal e
stim
atio
n, m
enta
l or w
ritt
en
stra
tegi
es, o
r by
usin
g a
calc
ulat
or
use
stra
tegi
es to
eva
luat
e th
e •
reas
onab
lene
ss o
f ans
wer
s
add
and
sub
trac
t fra
ctio
ns w
ith
•re
late
d d
enom
inat
ors
in re
al-li
fe
situ
atio
ns
add
and
sub
trac
t dec
imal
s in
real
-life
•
situ
atio
ns, i
nclu
din
g m
oney
esti
mat
e su
m, d
iffe
renc
e, p
rodu
ct
•an
d qu
otie
nt in
real
-life
sit
uati
ons,
in
clud
ing
frac
tion
s an
d d
ecim
als.
conv
ert i
mp
rop
er fr
acti
ons
to m
ixed
•
num
ber
s an
d vi
ce v
ersa
in re
al-li
fe
situ
atio
ns
sim
plif
y fr
acti
ons
in c
omp
utat
ion
•an
swer
s
use
frac
tion
s, d
ecim
als
and
•p
erce
ntag
es in
terc
hang
eab
ly in
real
-lif
e si
tuat
ions
sele
ct a
nd u
se a
n ap
pro
pri
ate
•se
quen
ce o
f op
erat
ions
to s
olve
wor
d p
rob
lem
s
sele
ct a
n ef
ficie
nt m
etho
d fo
r sol
ving
•
a p
rob
lem
: men
tal e
stim
atio
n, m
enta
l co
mp
utat
ion,
wri
tten
alg
orit
hms,
by
usin
g a
calc
ulat
or
use
stra
tegi
es to
eva
luat
e th
e •
reas
onab
lene
ss o
f ans
wer
s
use
men
tal a
nd w
ritt
en s
trat
egie
s fo
r •
add
ing,
sub
trac
ting
, mul
tip
lyin
g an
d d
ivid
ing
frac
tion
s an
d d
ecim
als
in
real
-life
sit
uati
ons
esti
mat
e an
d m
ake
app
roxi
mat
ions
in
•re
al-li
fe s
itua
tion
s in
volv
ing
frac
tion
s,
dec
imal
s an
d p
erce
ntag
es.
Learning continuums
Mathematics scope and sequence 25
No
tes
*To
cons
erve
, in
mat
hem
atic
al te
rms,
m
eans
the
amou
nt s
tays
the
sam
e re
gard
less
of t
he a
rran
gem
ent.
Lear
ners
who
hav
e b
een
enco
urag
ed
to s
elec
t the
ir ow
n ap
par
atus
and
m
etho
ds, a
nd w
ho b
ecom
e ac
cust
omed
to
dis
cuss
ing
and
ques
tion
ing
thei
r w
ork,
will
hav
e co
nfid
ence
in lo
okin
g fo
r al
tern
ativ
e ap
pro
ache
s w
hen
an in
itia
l at
tem
pt i
s un
succ
essf
ul.
Esti
mat
ion
is a
ski
ll th
at w
ill d
evel
op
wit
h ex
per
ienc
e an
d w
ill h
elp
chi
ldre
n ga
in a
“fe
el”
for n
umb
ers.
Chi
ldre
n m
ust
be
give
n th
e op
por
tuni
ty to
che
ck th
eir
esti
mat
es s
o th
at th
ey a
re a
ble
to fu
rthe
r re
fine
and
imp
rove
thei
r est
imat
ion
skill
s.
Ther
e ar
e m
any
opp
ortu
niti
es in
the
unit
s of
inqu
iry
and
duri
ng th
e sc
hool
day
for
stud
ents
to p
ract
ise
and
app
ly n
umb
er
conc
epts
aut
hent
ical
ly.
No
tes
**M
odel
ling
invo
lves
usi
ng c
oncr
ete
mat
eria
ls to
rep
rese
nt n
umb
ers
or
num
ber
op
erat
ions
, for
exa
mp
le, t
he u
se
of p
atte
rn b
lock
s or
frac
tion
pie
ces
to
rep
rese
nt fr
acti
ons
and
the
use
of b
ase
10
blo
cks
to re
pre
sent
num
ber
op
erat
ions
.
Stud
ents
nee
d to
use
num
ber
s in
m
any
situ
atio
ns in
ord
er to
ap
ply
thei
r un
der
stan
din
g to
new
sit
uati
ons.
In
add
itio
n to
the
unit
s of
inqu
iry,
chi
ldre
n’s
liter
atur
e al
so p
rovi
des
rich
op
por
tuni
ties
fo
r dev
elop
ing
num
ber
con
cep
ts.
To b
e us
eful
, ad
dit
ion
and
sub
trac
tion
fa
cts
need
to b
e re
calle
d au
tom
atic
ally
. Re
sear
ch c
lear
ly in
dic
ates
that
ther
e ar
e m
ore
effe
ctiv
e w
ays
to d
o th
is th
an “d
rill
and
pra
ctic
e”. A
bov
e al
l, it
hel
ps
to h
ave
stra
tegi
es fo
r wor
king
them
out
. Cou
ntin
g on
, usi
ng d
oub
les
and
usin
g 10
s ar
e g
ood
stra
tegi
es, a
ltho
ugh
lear
ners
freq
uent
ly
inve
nt m
etho
ds th
at w
ork
equa
lly w
ell f
or
them
selv
es.
Dif
ficul
ties
wit
h fr
acti
ons
can
aris
e w
hen
frac
tion
al n
otat
ion
is in
trod
uced
bef
ore
stud
ents
hav
e fu
lly c
onst
ruct
ed m
eani
ng
abou
t fra
ctio
n co
ncep
ts.
No
tes
Mod
ellin
g us
ing
man
ipul
ativ
es p
rovi
des
a
valu
able
sca
ffol
d fo
r con
stru
ctin
g m
eani
ng a
bou
t mat
hem
atic
al
conc
epts
. The
re s
houl
d b
e re
gula
r op
por
tuni
ties
for l
earn
ers
to w
ork
wit
h a
rang
e of
man
ipul
ativ
es a
nd to
d
iscu
ss a
nd n
egot
iate
thei
r dev
elop
ing
und
erst
and
ings
wit
h ot
hers
.
***E
xam
ple
s of
rela
ted
den
omin
ator
s in
clud
e ha
lves
, qua
rter
s (fo
urth
s) a
nd
eigh
ths.
The
se c
an b
e m
odel
led
easi
ly b
y fo
ldin
g st
rip
s or
squ
ares
of p
aper
.
The
inte
rpre
tati
on a
nd m
eani
ng o
f re
mai
nder
s ca
n ca
use
dif
ficul
ty fo
r so
me
lear
ners
. Thi
s is
esp
ecia
lly tr
ue if
ca
lcul
ator
s ar
e b
eing
use
d. F
or e
xam
ple
, 67
÷ 4
= 1
6.75
. Thi
s ca
n al
so b
e sh
own
as 1
6¾ o
r 16
r3. L
earn
ers
need
pra
ctic
e in
pro
duci
ng a
pp
rop
riat
e an
swer
s w
hen
usin
g re
mai
nder
s. F
or e
xam
ple
, for
a
scho
ol tr
ip w
ith
25 s
tud
ents
, onl
y b
uses
th
at c
arry
20
stud
ents
are
ava
ilab
le. A
re
mai
nder
cou
ld n
ot b
e le
ft b
ehin
d, s
o an
othe
r bus
wou
ld b
e re
quire
d!
Cal
cula
tor s
kills
mus
t not
be
igno
red
. A
ll an
swer
s sh
ould
be
chec
ked
for t
heir
reas
onab
lene
ss.
By re
flec
ting
on
and
reco
rdin
g th
eir
find
ings
in m
athe
mat
ics
lear
ning
logs
, st
uden
ts b
egin
to n
otic
e p
atte
rns
in th
e nu
mb
ers
that
will
furt
her d
evel
op th
eir
und
erst
and
ing.
No
tes
It is
not
pra
ctic
al to
con
tinu
e to
dev
elop
an
d us
e b
ase
10 m
ater
ials
bey
ond
1,00
0.
Lear
ners
sho
uld
have
litt
le d
iffic
ulty
in
exte
ndin
g th
e p
lace
val
ue s
yste
m o
nce
they
hav
e un
der
stoo
d th
e gr
oup
ing
pat
tern
up
to 1
,000
. The
re a
re a
num
ber
of
web
site
s w
here
vir
tual
man
ipul
ativ
es
can
be
utili
zed
for w
orki
ng w
ith
larg
er
num
ber
s.
Esti
mat
ion
pla
ys a
key
role
in c
heck
ing
the
feas
ibili
ty o
f ans
wer
s. T
he m
etho
d of
mul
tip
lyin
g nu
mb
ers
and
igno
ring
th
e d
ecim
al p
oint
, the
n ad
just
ing
the
answ
er b
y co
unti
ng d
ecim
al p
lace
s, d
oes
not g
ive
the
lear
ner a
n un
der
stan
din
g of
w
hy it
is d
one.
Ap
plic
atio
n of
pla
ce v
alue
kn
owle
dg
e m
ust p
rece
de
this
ap
plic
atio
n of
pat
tern
.
Mea
sure
men
t is
an e
xcel
lent
way
of
exp
lori
ng th
e us
e of
frac
tion
s an
d d
ecim
als
and
thei
r int
erch
ang
e.
Stud
ents
sho
uld
be
give
n m
any
opp
ortu
niti
es to
dis
cove
r the
link
b
etw
een
frac
tion
s an
d d
ivis
ion.
A th
orou
gh u
nder
stan
din
g of
m
ulti
plic
atio
n, fa
ctor
s an
d la
rge
num
ber
s is
requ
ired
bef
ore
wor
king
wit
h ex
pon
ents
.
Mathematics scope and sequence26
Samples
Several examples of how schools are using the planner to facilitate mathematics inquiries have been developed and trialled by IB World Schools offering the PYP. These examples are included in the HTML version of the mathematics scope and sequence on the online curriculum centre. The IB is interested in receiving planners that have been developed for mathematics inquiries or for units of inquiry where mathematics concepts are strongly evident. Please send planners to pyp@ibo.org for possible inclusion on this site.
Recommended